PHYSICAL REVIEW E 81, 016113 共2010兲
Global dynamic routing for scale-free networks Xiang Ling, Mao-Bin Hu,* Rui Jiang,† and Qing-Song Wu School of Engineering Science, University of Science and Technology of China, Hefei 230026, People’s Republic of China 共Received 30 July 2009; published 28 January 2010兲 Traffic is essential for many dynamic processes on networks. The efficient routing strategy 关G. Yan, T. Zhou, B. Hu, Z. Q. Fu, and B. H. Wang, Phys. Rev. E 73, 046108 共2006兲兴 can reach a very high capacity of more than ten times of that with shortest path strategy. In this paper, we propose a global dynamic routing strategy for network systems based on the information of the queue length of nodes. Under this routing strategy, the traffic capacity is further improved. With time delay of updating node queue lengths and the corresponding paths, the system capacity remains constant, while the travel time for packets increases. DOI: 10.1103/PhysRevE.81.016113
PACS number共s兲: 89.75.Hc, 89.20.Hh, 89.40.⫺a
I. INTRODUCTION
Dynamical properties of network traffic have attracted much attention from physical and engineering communities. The prototypes include the transfer of packets in the Internet, the flying of airplanes between airports, the motion of vehicles in urban network, the migration of carbon in biosystems, and so on. Since the discovery of small-world phenomenon 关1兴 and scale-free property 关2兴, it is widely proved that the topology and degree distribution of networks have profound effects on the processes taking place on these networks, including traffic flow 关3–6兴. The traffic jamming transition in the Internet was observed and analyzed in 1996 关7兴. In the past few years, the phase-transition phenomena 关8–11兴, the scaling of traffic fluctuations 关12–15兴, and the routing strategies 关16–33兴 of traffic dynamics have been widely studied. Compared with the high cost of changing the infrastructure, developing a better routing strategy is usually preferable to enhance the network capacity. Nowadays, the shortest path strategy is widely used in different systems. However, it often leads to the failure of hub routers with high degree and betweenness. In this light, some new routing strategies have been suggested, such as the efficient routing strategy 关21兴, the integration of shortest path and local information 关22兴, the nextnearest-neighbor searching strategy 关29兴, the local routing strategy 关31,32兴, and so on. In the efficient routing strategy 关21兴, the path between any nodes i 共source兲 and j 共destination兲 is defined as the path in which the sum degree of nodes is a minimum. It is denote as l
Pij = min 兺 k共xm兲 ,
共1兲
m=0
where k共xm兲 is the degree of node xm, l is the path length, and  is a tunable parameter. When  is set to 1, the largest network capacity can be achieved, where packets are propelled to the peripheral of the network. The efficient path can achieve a very high capacity for the network, which can be ten times of that with the shortest path strategy. In the local
*
[email protected] †
[email protected]
1539-3755/2010/81共1兲/016113共5兲
routing strategy 关31兴, the packets are forwarded by the local information of neighbors’ degree, ⌸l→i =
ki␣
兺j k␣j
共2兲
.
It is also proved that the best routing strategy emerges at ␣ = −1, where the packets are forced to select the low-degree nodes. In the routing strategy based on the integration of static and dynamic information 关32兴, the queue length of neighboring nodes is taken into consideration, ⌸l→i =
ki共ni + 1兲
兺j k j共n j + 1兲
,
共3兲
where ni is the queue length of the neighboring node i. It is shown that the hub nodes play an important role in the traffic process. In this paper, we propose that an efficient network routing strategy should not only consider the topology of the network, but also the effects of the queue length of nodes. We introduce a global dynamic routing strategy for the networks. In this routing strategy, the packets are delivered along the path in which the sum queue length of nodes is a minimum. Numerical results show that the system capacity can be further improved to almost two times of that with the efficient routing strategy 关21兴. The effects of the strategy on the efficiency of the scale-free traffic system are discussed in detail. The effects of the path update delay are also investigated. The paper is organized as follows. In the following section, we describe the network model and traffic rules in detail. In Sec. III, the simulation results are presented and discussed. In the last section, the work is concluded. II. TRAFFIC MODEL
Recent studies indicate that many communication systems such as the Internet and the World Wide Web are not homogeneous as random or regular networks, but heterogeneous with a degree distribution following the power-law distribution P共k兲 = k−␥. The Barabási-Albert 共BA兲 model 关2兴 is a well-known model which can generate networks with a power-law degree distribution. Without lose of generality, we
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PHYSICAL REVIEW E 81, 016113 共2010兲
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construct the underlying network structure by the BA network model. In this model, starting from m0 fully connected nodes, a new node with m edges 共m ⱕ m0兲 is added to the existing graph at each time step according to preferential attachment, i.e., the probability ⌸i of being connected to the existing node i is proportional to the degree ki of the node, ⌸i =
ki
兺j k j
.
共4兲
The traffic model is described as follows: at each time step, R packets enter the system with randomly chosen sources and destinations. One node can deliver at most C 共here, we set C = 1.0 for each node兲 packets to its neighboring nodes. From the many paths between the source and destination, we selected the path in which the sum of node queue lengths is a minimum. Therefore, the path between nodes i 共source兲 and j 共destination兲 can be denoted as l
Pij = min 兺 关1 + n共xm兲兴,
共5兲
m=0
where n共xm兲 is the queue length of node xm and l is the path length. The maximal queue length of each node is assumed to be unlimited and the first-in-fist-out discipline is applied at each queue. Once a packet arrives at its destination, it is removed from the system. In order to describe the phase transition of traffic flow in the network, we use the order parameter 关8兴 C 具⌬N p典 , t→⬁ R ⌬t
共R兲 = lim
共6兲
where ⌬N p = N p共t + ⌬t兲 − N p共t兲, 具 ¯ 典 indicates the average over time windows of width ⌬t, and N p共t兲 is the total number of packets within the network at time t. The order parameter represents the balance between the inflow and outflow of packets. In the free flow state, due to the balance of created and removed packets, is around zero. With increasing packet generation rate R, there will be a critical value of Rc that characterizes the phase transition from free flow to congestion. When R exceeds Rc, the packets accumulate continuously in the network, and will become a constant larger than zero. The network capacity can be measured by the maximal generating rate Rc at the phase-transition point.
FIG. 1. 共Color online兲 Evolution of packet number in the network under different routing strategies. 共a兲 Shortest path routing strategy, 共b兲 efficient routing strategy, and 共c兲 global dynamic routing strategy. 共d兲 The order parameter vs R under the three routing strategies.
In Fig. 2, we show the relation of network’s capacity Rc vs the average degree 具k典, and vs the network size N under three routing strategies. Figure 2共a兲 shows that with the same network size, the network’s capacity increases with the average degree 具k典 under three routing strategies. The rank of network capacities is global dynamic routing ⬎ efficient routing⬎ shortest path routing. In Fig. 2共b兲, one can also see that the network’s capacity increases with the increasing of the network size N. Again, with the same average degree or network size, the network’s capacity under the global dynamic routing strategy is the largest. Because the node queue length changes from time to time, it is computation consuming to find the global dynamic paths in each step. Therefore, it is important to introduce a time delay 共␦T兲 for the update of the global queue information and the corresponding paths. Figure 3 shows the evolution of the total packet number N p for different R with a time delay of ␦T = 20. Although the dynamic paths are updated every 20 time steps, the capacity of network still remains at Rc = 41. Figure 4 shows the evolution of the total packet number N p for other values of ␦T. One can see that the capacity of network under different ␦T
III. SIMULATION RESULTS
Figures 1共a兲–1共c兲 show the evolution of the total packet number N p under shortest, efficient, and global dynamic paths with the network size N = 500 and average degree 具k典 = 4. Figure 1共d兲 compares the relation of order parameter vs R under the three routing strategies. One can see that the network capacity under the shortest path routing strategy is Rc = 3. The efficient routing strategy has Rc = 20. The global dynamic routing strategy reaches a very high capacity of Rc = 41, which is more than double of the efficient routing strategy. As far as we know, the global dynamic routing strategy can achieve the highest traffic capacity.
FIG. 2. 共Color online兲 共a兲 The network capacity Rc vs average degree 具k典 with the same network size N = 500 under the three different routing strategies. 共b兲 The network capacity Rc vs network size N with the same average degree 具k典 = 4 under the three routing strategies.
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FIG. 3. 共Color online兲 Evolution of packet number N p for different R with time delay of ␦T = 20. The inset depicts the order parameter vs R with ␦T = 20. The network capacity remains at Rc = 41. Other parameters are network size N = 500 and average degree 具k典 = 4.
remains the same 共Rc = 41兲. This demonstrates that the capacity of network is independent of ␦T. However, one can see that the saturate value of N p increased with the increase in ␦T, as shown in the inset of Fig. 4. This indicates that the queue length of nodes in the system will increase, although the network capacity remains constant at Rc = 41. In Fig. 4, one can see that N p first increase to a maximum and then decrease to a balanced state. This phenomenon is more obvious with larger values of ␦T. This can be explained as follows. At the beginning of transportation, there is no packets in each node 关n共xm兲 = 0兴. Thus, the packets are delivered following the shortest path routing strategy until the time reaches T = ␦T. Because the network capacity of the shortest path routing is only Rc = 3, N p will increase rapidly as in the congested state. When T = ␦T, the global dynamic routing strategy begins to take action, so that N p decreases to a balanced state. Figure 5 shows the dependence of the average queue length n共k兲 on the degree k with different ␦T in the free flow state. For different ␦T, n共k兲 follows a power law of n共k兲
FIG. 4. 共Color online兲 Evolution of N p for different time delay with R = Rc = 41. For all ␦T, the network capacity remains at Rc = 41. The inset depicts the saturate value of N p at the balanced period vs ␦T under R = Rc = 41. Other parameters are network size N = 500 and average degree 具k典 = 4.
FIG. 5. 共Color online兲 Average packet number in nodes n共k兲 vs node degree k with different ␦T. Network size N = 500 and average degree 具k典 = 4.
⬃ k␥ with exponent ␥ ⬎ 0. This indicates that the hub nodes are more burdened with this routing strategy. Moreover, for each k, the value of n共k兲 increases with ␦T. This is in agreement with that the total packet number increased with ␦T under the same R 共Fig. 4, inset兲. Then we investigated the probability distribution of the packet travel time for different ␦T in the free flow state. Packet’s travel time is an important factor for characterizing the network’s behavior. The travel time is the time that the packet spends traveling from the source to destination. In Fig. 6, one can see that the probability of traveling time approximately follows a Poisson distribution under different ␦T. The travel time corresponding to the peak of distribution will increase with ␦T. This indicates that the packets need to travel for longer times if the global dynamic paths are updated with the time delay. Figure 7 shows the average travel time and the waiting time for different ␦T. The waiting time is the time that the packets spend in a queue waiting for the delivery. One can see that both the average travel time and the average waiting time increase almost linearly with ␦T. This phenomenon is consistent with the result of Fig. 6. With the same R, when ␦T increases, the queue length in each node increases 共see Fig. 5兲, so the waiting time of packets also increases. As a
FIG. 6. 共Color online兲 The probability distribution of travel time for different ␦T. Network parameters are N = 500 and average degree 具k典 = 4.
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FIG. 8. 共Color online兲 The probability distribution of path length for different ␦T with network size N = 500 and average degree 具k典 = 4. The inset depicts the average value of path length vs ␦T.
FIG. 7. 共a兲 Average travel time vs ␦T with network size N = 500 and average degree 具k典 = 4. 共b兲 Average waiting time vs ␦T with network size N = 500 and average degree 具k典 = 4.
result, the average traveling time will also increase. This indicates that with the introduction of the time delay, the system’s capacity remains the same at the cost of increasing the packets’ travel time. Finally, we investigate the probability distribution of the path length for different ␦T in the free flow state. The path length is defined as the number of nodes that the packet travels from the source to destination. As shown in Fig. 8, the probability of the path length approximately follows a Poisson distribution for different ␦T. From the inset of Fig. 8, one can also see that the average path lengths increase very slightly with ␦T. Therefore, the increment of packets’ travel time comes mainly from the increment of the waiting time in each queue along the path, but not from the increment of the path length.
packet to use the peripheral nodes of the network. The global dynamic routing strategy still encourages us to use the hub nodes 共see Fig. 5兲, but it can achieve a much higher capacity for the system. We also investigate the delay effects of the path update. With the increment of the update delay, the network’s traffic capacity remains the same, but the total packet number in system, the traveling time, and the waiting time will increase. The delay of the path update will not affect the overall capacity and the average path length. The increment of the travel time comes mainly from the increase in the waiting time in the queues. Since the traffic process is crucial for many industrial and communication systems, the global dynamic strategy can be useful for the design and optimization of routing strategies for these systems, including the Internet, the World Wide Web, the urban transportation system 关34–36兴, the power grid, the airway system, and so on. The study can also be interesting from a theoretical point of view, as an upper bound for the efficiency. In the future work, we will apply the global dynamic routing strategy in other network structures. It is expected the traffic capacity of these networks will be enhanced with this routing strategy.
IV. CONCLUSION
ACKNOWLEDGMENTS
In summary, we propose a global dynamic routing strategy for scale-free networks. Under this strategy, the traffic capacity is much larger than other previously known routing strategies. Compared with the efficient routing strategy 关21兴 and the local routing strategy 关31兴, the global dynamic routing strategy differs from their mechanism of propelling the
This work was funded by National Basic Research Program of China 共Contract No. 2006CB705500兲; the NNSFC Key Project No. 10532060; NNSFC Projects No. 70601026, No. 10672160, and No. 10872194; and the Innovation Foundation for Graduate Students of University of Science and Technology of China 共Contract No. W21103010012兲.
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